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Linear Algebra in Action

Harry Dym

Chapter 5

Determinants - all with Video Answers

Educators

Chapter Questions

06:09

Problem 1

Show that if $P$ is a simple permutation, then $P=P^T$.

Sirat Shah
Sirat Shah
Numerade Educator

Problem 2

Use the three rules in Theorem 5.1 to show that if $A \in \mathbb{C}^{2 \times 2}$, then $\operatorname{det} A=a_{11} a_{22}-a_{12} a_{21}$.

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05:03

Problem 3

Use the three rules in Theorem 5.1 to show that if $A \in \mathbb{C}^{3 \times 3}$, then $\operatorname{det} A=a_{11} a_{22} a_{33}-a_{11} a_{23} a_{32}+a_{12} a_{23} a_{31}-a_{12} a_{21} a_{33}+a_{13} a_{21} a_{32}-a_{13} a_{22} a_{31}$.

Suman Saurav Thakur
Suman Saurav Thakur
Numerade Educator
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Problem 4

Let
$$
E=\left[\begin{array}{cccc}
1 & 0 & 0 & 0 \\
\alpha_{21} & 1 & 0 & 0 \\
\alpha_{31} & \alpha_{32} & 1 & 0 \\
\alpha_{41} & \alpha_{42} & \alpha_{43} & 1
\end{array}\right]
$$
and let $\mathbf{e}_i, i=1, \ldots, 4$ denote the standard basis for $\mathbb{C}^4$. Show that
$$
\begin{aligned}
E= & \left(I_4+\alpha_{21} \mathbf{e}_2 \mathbf{e}_1^T\right)\left(I_4+\alpha_{31} \mathbf{e}_3 \mathbf{e}_1^T\right)\left(I_4+\alpha_{41} \mathbf{e}_4 \mathbf{e}_1^T\right) \\
& \times\left(I_4+\alpha_{32} \mathbf{e}_3 \mathbf{e}_2^T\right)\left(I_4+\alpha_{42} \mathbf{e}_4 \mathbf{e}_2^T\right)\left(I_4+\alpha_{43} \mathbf{e}_4 \mathbf{e}_3^T\right) .
\end{aligned}
$$
$\mathbf{9}^{\circ}$ If $A \in \mathbb{C}^{n \times n}$, then $A$ is invertible if and only if $\operatorname{det}(A) \neq 0$.

Nick Johnson
Nick Johnson
Numerade Educator
01:39

Problem 5

Show that if $\operatorname{det} B \neq 0$, then the functional $\varphi(A)=\frac{\operatorname{det}(A B)}{\operatorname{det} B}$ meets conditions $1^{\circ}-3^{\circ}$.

Raj Bala
Raj Bala
Numerade Educator

Problem 6

Complete the proof of $13^{\circ}$.

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07:37

Problem 7

Calculate the determinants of the following matrices by Gaussian elimination:
$$
\left[\begin{array}{llll}
1 & 3 & 2 & 1 \\
0 & 4 & 1 & 6 \\
0 & 0 & 2 & 1 \\
1 & 1 & 0 & 4
\end{array}\right],\left[\begin{array}{llll}
1 & 0 & 1 & 0 \\
0 & 1 & 0 & 1 \\
1 & 0 & 0 & 1 \\
0 & 1 & 1 & 0
\end{array}\right],\left[\begin{array}{llll}
1 & 3 & 2 & 4 \\
0 & 2 & 1 & 6 \\
0 & 0 & 3 & 0 \\
0 & 0 & 1 & 2
\end{array}\right],\left[\begin{array}{llll}
0 & 0 & 0 & 4 \\
1 & 2 & 3 & 1 \\
0 & 0 & 1 & 1 \\
0 & 1 & 2 & 6
\end{array}\right] .
$$

Sirat Shah
Sirat Shah
Numerade Educator
01:13

Problem 8

Calculate the determinants of the matrices in the previous exercise by rules $1^{\circ}$ to $13^{\circ}$.

Harmender Singh Yadav
Harmender Singh Yadav
Numerade Educator

Problem 9

Show that if $J=\operatorname{diag}\left\{C_{\lambda_1}^{(5)}, C_{\lambda_2}^{(3)}, C_{\lambda_3}^{(2)}\right\}$, then $\left(J-\lambda_1 I_{10}\right)^5\left(J-\lambda_2 I_{10}\right)^3\left(J-\lambda_3 I_{10}\right)^2=O$.

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Problem 10

Show that if $\lambda_1=\lambda_2$ in Exercise 5.9, then $\left(J-\lambda_1 I_{10}\right)^5\left(J-\lambda_3 I_{10}\right)^2=O$.

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Problem 11

Show that if $A, B \in \mathbb{C}^{n \times n}$ are expressed in compatible four block form with $A_{11}, B_{11} \in \mathbb{C}^{k \times k}, A_{22}, B_{22} \in \mathbb{C}^{(n-k) \times(n-k)}$ and if $A B=I_n$, then
$\operatorname{det} A_{11} \neq 0 \Longleftrightarrow \operatorname{det} B_{22} \neq 0$ and $\operatorname{det} A_{22} \neq 0 \Longleftrightarrow \operatorname{det} B_{11} \neq 0$.

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Problem 12

Show that if $A, B \in \mathbb{C}^{n \times n}$ are expressed in compatible four block form with $A_{11}, B_{11} \in \mathbb{C}^{k \times k}, A_{22}, B_{22} \in \mathbb{C}^{(n-k) \times(n-k)}$ and if $A B=I_n$, then
$$
\operatorname{det} B_{22}=\frac{\operatorname{det} A_{11}}{\operatorname{det} A} \quad \text { and } \quad \operatorname{det} B_{22}=\frac{\operatorname{det} A_{22}}{\operatorname{det} A} \text {. }
$$

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Problem 13

Show that if $B$ is a $p \times q$ matrix and $C$ is a $q \times p$ matrix, then
$$
\operatorname{det}\left\{I_p-B C\right\}=\operatorname{det}\left\{I_q-C B\right\}
$$
and that
$$
q+\operatorname{rank}\left\{I_p-B C\right\}=p+\operatorname{rank}\left\{I_q-C B\right\} .
$$

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Problem 14

Show that if $A \in \mathbb{C}^{p \times q}$ and $B \in \mathbb{C}^{q \times p}$, then
$$
\operatorname{det}\left(\lambda I_p-A B\right)=\lambda^{p-q} \operatorname{det}\left(\lambda I_q-B A\right) \text { if } \lambda \neq 0 .
$$

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02:13

Problem 15

Show that if $\mathbf{u} \in \mathbb{C}^p$, then $\operatorname{det}\left(I_p-\mathbf{u u}^H\right) \neq 0$ if and only if $\mathbf{u}^H \mathbf{u} \neq 1$.

Victor Salazar
Victor Salazar
Numerade Educator
04:09

Problem 16

Let $A \in \mathbb{C}^{n \times n}$ be invertible and let $\mathbf{u}, \mathbf{v} \in \mathbb{C}^n$. Show that the matrix $A+\mathbf{u v}^T$ is invertible if and only if $1+\mathbf{v}^T A^{-1} \mathbf{u} \neq 0$.

Anthony Ramos
Anthony Ramos
Numerade Educator
04:50

Problem 17

Calculate the determinant of the matrix
$$
A=\left[\begin{array}{lllllll}
1 & 1 & 1 & 1 & 0 & 0 & 0 \\
0 & 2 & 2 & 2 & 0 & 0 & 0 \\
0 & 0 & 3 & 3 & 0 & 0 & 0 \\
0 & 0 & 0 & 4 & 0 & 0 & 0 \\
9 & 8 & 7 & 6 & 1 & 2 & 3 \\
1 & 5 & 9 & 3 & 0 & 4 & 1 \\
8 & 8 & 8 & 6 & 0 & 2 & 2
\end{array}\right]
$$

Melissa Munoz
Melissa Munoz
Numerade Educator
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Problem 18

Calculate the determinant of the matrix $\left[\begin{array}{cc}O & I_n \\ I_k & O\end{array}\right]$.

Nick Johnson
Nick Johnson
Numerade Educator

Problem 19

Verify formulas (5.19) and (5.20).

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05:02

Problem 20

Write the Binet-Cauchy formula for $\operatorname{det} A B$ for $A \in \mathbb{C}^{2 \times 3}$ and $B \in \mathbb{C}^{2 \times 3}$ in terms of the entries $a_{i j}$ of $A$ and $b_{i j}$ of $B$. Do not compute the relevant determinants.

Anthony Ramos
Anthony Ramos
Numerade Educator
03:37

Problem 21

Show that if $A, B \in \mathbb{C}^{5 \times 5}$ and $A B=I_5$, then
$$
B\left(\begin{array}{lll}
2 & 3 & 5 \\
1 & 4 & 5
\end{array}\right)=\frac{A\left(\begin{array}{ll}
1 & 4 \\
2 & 3
\end{array}\right)}{\operatorname{det} A} \text { and } B\left(\begin{array}{ll}
2 & 5 \\
1 & 3
\end{array}\right)=-\frac{A\left(\begin{array}{lll}
1 & 3 & 4 \\
2 & 4 & 5
\end{array}\right)}{\operatorname{det} A} \text {. }
$$

Manisha Sarker
Manisha Sarker
Numerade Educator
01:50

Problem 22

Evaluate the determinant of the $4 \times 4$ matrix
$$
A=\left[\begin{array}{llll}
5 & 2 & 3 & 1 \\
3 & 0 & 0 & 2 \\
1 & 1 & 0 & 1 \\
0 & 2 & 0 & 1
\end{array}\right]
$$
twice, first begin by expanding in minors along the third column and then begin by expanding in minors along the fourth column.

Lindsay El
Lindsay El
Numerade Educator
02:36

Problem 23

Show that if $A \in \mathbb{C}^{n \times n}$, then the $i j$ minor $A_{\{i j\}}$ is equal to $(-1)^{i+j} \operatorname{det} \tilde{A}$, where $\tilde{A}$ denotes the matrix $A$ with its $i$ 'th row replaced by $\mathbf{e}_j^T$

James Kiss
James Kiss
Numerade Educator

Problem 24

Formulate and verify the analogue of formula (5.24) for $4 \times 4$ matrices.

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Problem 25

Let $A$ be an $n \times n$ matrix and let $f(x)=\operatorname{det}\left(I_n+x A\right)$. Show that $f^{\prime}(0)=\operatorname{trace} A$.

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Problem 26

Give a second proof of the formula in Exercise 5.25 on the basis of the Jordan decomposition $A=U J U^{-1}$ that is described in Theorem 4.14.

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03:59

Problem 27

Show that if $A \in \mathbb{C}^{3 \times 3}$ is invertible and $A \mathbf{x}=\mathrm{b}$, then
$$
x_1=\frac{\operatorname{det}\left[\begin{array}{lll}
b_1 & a_{12} & a_{13} \\
b_2 & a_{22} & a_{23} \\
b_3 & a_{32} & a_{33}
\end{array}\right]}{\operatorname{det} A}, x_2=\frac{\operatorname{det}\left[\begin{array}{lll}
a_{11} & b_1 & a_{13} \\
a_{21} & b_2 & a_{23} \\
a_{31} & b_3 & a_{33}
\end{array}\right]}{\operatorname{det} A}
$$
and state and verify the analogous formula for $x_3$. [REMARK: This is an example of Cramer's rule.]

Victor Salazar
Victor Salazar
Numerade Educator
01:41

Problem 28

Show that if $A, B \in \mathbb{C}^{n \times n}$ and $A B=I_n$, then for every $\lambda \in \mathbb{C}$
$$
b_{11}+b_{21} \lambda+\cdots+b_{n 1} \lambda^{n-1}=\frac{\operatorname{det}\left[\begin{array}{cccc}
1 & \lambda & \cdots & \lambda^{n-1} \\
a_{21} & a_{22} & \cdots & a_{2 n} \\
\vdots & & & \vdots \\
a_{n 1} & a_{n 2} & \cdots & a_{2 n n}
\end{array}\right]}{\operatorname{det} A} .
$$

Adhish Rele
Adhish Rele
Numerade Educator
02:33

Problem 29

Compute the inverse of the matrix $A=\left[\begin{array}{lll}1 & 2 & 2 \\ 2 & 1 & 2 \\ 1 & x & 0\end{array}\right]$ for those values of $x$ for which $A$ is invertible. [HINT: Exploit formula (5.24).]

Shahab Ullah
Shahab Ullah
Numerade Educator
08:08

Problem 30

Show that if
$$
\operatorname{det}\left[\begin{array}{ll}
a_{11} & a_{12} \\
a_{21} & a_{22}
\end{array}\right] \alpha+\operatorname{det}\left[\begin{array}{ll}
a_{11} & a_{13} \\
a_{21} & a_{23}
\end{array}\right] \beta+\operatorname{det}\left[\begin{array}{ll}
a_{12} & a_{13} \\
a_{22} & a_{23}
\end{array}\right] \gamma=1 \text {. }
$$
then there exists a matrix $C \in \mathbb{C}^{3 \times 2}$ such that $\left[\begin{array}{lll}a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23}\end{array}\right] C=I_2$.

Lucas Finney
Lucas Finney
Numerade Educator
06:09

Problem 31

Let $\mathbf{a}_i \in \mathbb{C}^2$ and $\alpha_{i j}=\operatorname{det}\left[\begin{array}{ll}\mathbf{a}_i & \mathbf{a}_j\end{array}\right]$ for $i, j=1, \ldots, 4$. Show that
$$
\alpha_{12} \alpha_{34}-\alpha_{13} \alpha_{24}+\alpha_{14} \alpha_{23}=0 .
$$

Suman Saurav Thakur
Suman Saurav Thakur
Numerade Educator
01:02

Problem 32

Show that if the six numbers $\alpha_{12}, \alpha_{13}, \alpha_{14}, \alpha_{34}, \alpha_{24}$ and $\alpha_{23}$ satisfy the identity (5.27), then there exists a matrix $A \in \mathbb{C}^{2 \times 4}$ such that $\alpha_{i j}=\operatorname{det}\left[\begin{array}{ll}\mathbf{a}_i & \mathbf{a}_j\end{array}\right]$, where $\mathbf{a}_i$ designates the $i^{\prime}$ 'th column of $A$.

Raj Bala
Raj Bala
Numerade Educator
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Problem 33

Let $A \in \mathbb{C}^{p \times q}$. Show that $\operatorname{rank} A=r$ if and only if the largest square invertible submatrix of $A$ is of size $r \times r$. [REMARK: A $k \times k$ submatrix of $A$ is obtained by deleting $p-k$ rows and $q-k$ columns.]

Victor Salazar
Victor Salazar
Numerade Educator
01:12

Problem 34

Verify the formula in (1) for $n=2$ and $n=3$.

Carson Merrill
Carson Merrill
Numerade Educator
08:45

Problem 35

Verify the formula in (1) for an arbitrary positive integer $n$ by induction. [HINT: Expand in minors along the last column.]

Steven Clarke
Steven Clarke
Numerade Educator

Problem 36

Verify assertion (4) in Theorem 5.11.

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07:43

Problem 37

Show that if $p(\lambda)=a_0+a_1 \lambda+a_2 \lambda^2+\lambda^3=(\lambda-\mu)^3$, then
$$
\left[\begin{array}{ccc}
0 & 1 & 0 \\
0 & 0 & 1 \\
-a_0 & -a_1 & -a_2
\end{array}\right] V=V\left[\begin{array}{lll}
\mu & 1 & 0 \\
0 & \mu & 1 \\
0 & 0 & \mu
\end{array}\right] \text {, where } V=\left[\begin{array}{ccc}
1 & 0 & 0 \\
\mu & 1 & 0 \\
\mu^2 & 2 \mu & 1
\end{array}\right]
$$

Anthony Ramos
Anthony Ramos
Numerade Educator
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Problem 38

Show that if $P$ denotes the permutation $P$ defined by formula (5.30), then $\operatorname{det}\left(\lambda I_n-P\right)=-1+\lambda^n$ and $P^n=I_n$. [HINT: Invoke Theorem 5.11 and the Cayley-Hamilton theorem.]

Michelle Z.
Michelle Z.
Numerade Educator
21:28

Problem 39

Let $A \in \mathbb{C}^{n \times n}$ denote the circulant matrix defined by formula (5.29) and let $p(\lambda)=a_0+a_1 \lambda+\cdots+a_{n-1} \lambda^{n-1}$. Show that
$$
A V=V D, \quad \text { where } \quad D=\operatorname{diag}\left\{p\left(\zeta_1\right), \ldots, p\left(\zeta_n\right)\right\},
$$
$\zeta_j=\exp (2 \pi i j / n)$ for $j=1, \ldots, n$ and $V$ is a Vandermonde matrix with $\lambda_j=\zeta_j$ for $j=1, \ldots, n$. Conclude that
$$
\operatorname{det} A=p\left(\zeta_1\right) \cdots p\left(\zeta_n\right) .
$$

R M
R M
Numerade Educator
01:01

Problem 40

Show that if $A \in \mathbb{C}^{n \times n}$ and $B \in \mathbb{C}^{n \times n}$ are circulants, then $A B=B A$.

Nick Johnson
Nick Johnson
Numerade Educator

Problem 41

Find the determinant of the (Vandermonde) matrix $V$ given by formula (5.31) when $n=3$. [REMARK: You can calculate this determinant by brute force. But a better way is to let $f(x)$ denote the value of the determinant when $\lambda_1$ is replaced by $x$ and observe that $f(x)$ is a polynomial of degree two such that $f\left(\lambda_2\right)=f\left(\lambda_3\right)=0$.]

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Problem 42

Let $\left\{\alpha_0, \ldots, \alpha_n\right\}$ and $\left\{\beta_0, \ldots, \beta_n\right\}$ be two sets of points in $\mathbb{C}$. Show that if $\alpha_i \neq \alpha_j$ when $i \neq j$, then there exists a unique polynomial $p(\lambda)=c_0+c_1 \lambda+\cdots+c_n \lambda^n$ of degree $n$ such that $p\left(\alpha_i\right)=\beta_i$ for $i=0, \ldots, n$ and find a formula for the coefficients $c_j$.

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